# A criterion related to the Riemann Hypothesis

## TL;DR

This paper explores a specific distance measure related to the Riemann Hypothesis within the Nyman-Beurling-Báez-Duarte framework, demonstrating how the presence of zeros off the critical line affects this criterion.

## Contribution

It introduces a new criterion based on the distance measure that indicates the non-validity of the Riemann Hypothesis if zeros off the critical line exist.

## Key findings

- Shows how zeros off the critical line influence the distance measure
- Provides a criterion for disproving the Riemann Hypothesis
- Connects zero distribution to functional approximation in the Nyman-Beurling approach

## Abstract

A crucial role in the Nyman-Beurling-B\'aez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty\left|1-\zeta A_N\left(\frac{1}{2}+it\right)\right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] where the infimum is over all Dirichlet polynomials $$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$ of length $N$. In this paper we investigate $d_N^2$ under the assumption that the Riemann zeta function has four non-trivial zeros off the critical line. Thus we obtain a criterion for the non validity of the Riemann Hypothesis.